U.S. patent number 7,251,638 [Application Number 10/792,292] was granted by the patent office on 2007-07-31 for intelligent robust control system for motorcycle using soft computing optimizer.
This patent grant is currently assigned to Yamaha Hatsudoki Kabushiki Kaisha. Invention is credited to Shigeru Fujii, Sergey A. Panfilov, Kazuki Takahashi, Sergey V. Ulyanov, Hitoshi Watanabe.
United States Patent |
7,251,638 |
Fujii , et al. |
July 31, 2007 |
**Please see images for:
( Certificate of Correction ) ** |
Intelligent robust control system for motorcycle using soft
computing optimizer
Abstract
A Soft Computing (SC) optimizer for designing a Knowledge Base
(KB) to be used in a control system for controlling a motorcycle is
described. In one embodiment, a simulation model of the motorcycle
and rider control is used. In one embodiment, the simulation model
includes a feedforward rider model. The SC optimizer includes a
fuzzy inference engine based on a Fuzzy Neural Network (FNN). The
SC Optimizer provides Fuzzy Inference System (FIS) structure
selection, FIS structure optimization method selection, and
teaching signal selection and generation. The user selects a fuzzy
model, including one or more of: the number of input and/or output
variables; the type of fuzzy inference; and the preliminary type of
membership functions. A Genetic Algorithm (GA) is used to optimize
linguistic variable parameters and the input-output training
patterns. A GA is also used to optimize the rule base, using the
fuzzy model, optimal linguistic variable parameters, and a teaching
signal. The GA produces a near-optimal FNN. The near-optimal FNN
can be improved using classical derivative-based optimization
procedures. The FIS structure found by the GA is optimized with a
fitness function based on a response of the actual plant model of
the controlled plant. The SC optimizer produces a robust KB that is
typically smaller that the KB produced by prior art methods.
Inventors: |
Fujii; Shigeru (Iwata,
JP), Watanabe; Hitoshi (Iwata, JP),
Panfilov; Sergey A. (Crema, IT), Takahashi;
Kazuki (Crema, IT), Ulyanov; Sergey V. (Crema,
IT) |
Assignee: |
Yamaha Hatsudoki Kabushiki
Kaisha (JP)
|
Family
ID: |
34750598 |
Appl.
No.: |
10/792,292 |
Filed: |
March 3, 2004 |
Prior Publication Data
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Document
Identifier |
Publication Date |
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US 20050197994 A1 |
Sep 8, 2005 |
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Current U.S.
Class: |
706/23; 700/28;
700/50; 706/1; 706/19 |
Current CPC
Class: |
B62K
21/00 (20130101) |
Current International
Class: |
G05B
13/00 (20060101); G05B 13/04 (20060101); G05B
15/00 (20060101) |
Field of
Search: |
;706/59,8,50,1,23,19
;700/28,50 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
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1 085 460 |
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Mar 2001 |
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EP |
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1083520 |
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Mar 2001 |
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EP |
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10268904 |
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Oct 1998 |
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JP |
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WO 01-67186 |
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Sep 2001 |
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WO |
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Primary Examiner: Knight; Anthony
Assistant Examiner: Kennedy; Adrian L.
Attorney, Agent or Firm: Knobbe, Martens, Olson & Bear,
LLP
Claims
The invention claimed is:
1. A method for optimizing a knowledge base in a soft computing
controller for maneuvering a motorcycle, comprising: selecting a
fuzzy model by selecting one or more parameters, said one or more
parameters comprising at least one of a number of input variables,
a number of output variables, a type of fuzzy inference model, and
a teaching signal; optimizing linguistic variable parameters of a
knowledge base according to said one or more parameters to produce
optimized linguistic variables according to a teaching signal
obtained from a dynamic simulation model of a motorcycle and rider;
ranking rules in said rule base according to firing strength; and
eliminating rules with relatively weak firing strength leaving
selected rules from said rules in said rule base; optimizing said
selected rules, using said fuzzy model, said linguistic variable
parameters and said optimized linguistic variables, to produce
optimized selected rules.
2. The method of claim 1, further comprising optimizing said
selected rules using a derivative-based optimization procedure.
3. The method of claim 1, further comprising optimizing parameters
of membership functions of said optimized selected rules to reduce
approximation errors.
Description
BACKGROUND
1. Field of the Invention
The present invention relates to electronically-controlled
motorcycle steering and maneuverability systems based on soft
computing.
2. Description of the Related Art
Feedback control systems are widely used to maintain the output of
a dynamic system at a desired value in spite of external
disturbances that would displace it from the desired value. For
example, a household space-heating furnace, controlled by a
thermostat, is an example of a feedback control system. The
thermostat continuously measures the air temperature inside the
house, and when the temperature falls below a desired minimum
temperature the thermostat turns the furnace on. When the interior
temperature reaches the desired minimum temperature, the thermostat
turns the furnace off. The thermostat-furnace system maintains the
household temperature at a substantially constant value in spite of
external disturbances such as a drop in the outside temperature.
Similar types of feedback controls are used in many
applications.
A central component in a feedback control system is a controlled
object, a machine or a process that can be defined as a "plant",
whose output variable is to be controlled. In the above example,
the "plant" is the house, the output variable is the interior air
temperature in the house and the disturbance is the flow of heat
(dispersion) through the walls of the house. The plant is
controlled by a control system. In the above example, the control
system is the thermostat in combination with the furnace. The
thermostat-furnace system uses simple on-off feedback control
system to maintain the temperature of the house. In many control
environments, such as motor shaft position or motor speed control
systems, simple on-off feedback control is insufficient. More
advanced control systems rely on combinations of proportional
feedback control, integral feedback control, and derivative
feedback control. A feedback control based on a sum of
proportional, plus integral, plus derivative feedback is often
referred as a linear control. Similarly, Proportional feedback
control is often referred to as P control, and Proportional plus
Derivative feedback is often referred to as P(D) control, and
Proportional plus Integral feedback is referred as P(I)
control.
A linear control system (e.g., P, P(I), P(D), P(I)D, etc.) is based
on a linear model of the plant. In classical control systems, a
linear model is obtained in the form of ordinary differential
equations. The plant is assumed to be relatively linear, time
invariant, and stable. However, many real-world plants are time
varying, highly non-linear, and unstable. For example, the dynamic
model may contain parameters (e.g., masses, inductance,
aerodynamics coefficients, etc.), which are either only
approximately known or depend on a changing environment. If the
parameter variation is small and the dynamic model is stable, then
the linear controller may be satisfactory. However, if the
parameter variation is large or if the dynamic model is unstable,
then it is common to add Adaptive or Intelligent (AI) control
functions to the linear control system.
AI control systems use an optimizer, typically a non-linear
optimizer, to program the operation of the linear controller and
thereby improve the overall operation of the control system.
Classical advanced control theory is based on the assumption that
all controlled "plants" can be approximated as linear systems near
equilibrium points. Unfortunately, this assumption is rarely true
in the real world. Most plants are highly nonlinear, and often do
not have simple control algorithms. In order to meet these needs
for a nonlinear control, systems have been developed that use Soft
Computing (SC) concepts such Fuzzy Neural Networks (FNN), Fuzzy
Controllers (FC), and the like. By these techniques, the control
system evolves (changes) in time to adapt itself to changes that
may occur in the controlled "plant" and/or in the operating
environment.
Control systems based on SC typically use a Knowledge Base (KB) to
contain the knowledge of the FC system. The KB typically has many
rules that describe how the SC determines control parameters during
operation. Thus, the performance of an SC controller depends on the
quality of the KB and the knowledge represented by the KB.
Increasing the number of rules in the KB generally increases (very
often with redundancy) the knowledge represented by the KB but at a
cost of more storage and more computational complexity. Thus,
design of a SC system typically involves tradeoffs regarding the
size of the KB, the number of rules, the types of rules. etc.
Unfortunately, the prior art methods for selecting KB parameters
such as the number and types of rules are based on ad hoc
procedures using intuition and trial-and-error approaches.
Steering and/or maneuverability control of a motorcycle using soft
computing is particularly difficult because of the difficulty in
obtaining a sufficiently optimal knowledge base (KB). If the KB
does not contain enough knowledge about the dynamics of the
motorcycle-rider system, then the soft computing controller will
not be able to steer and/or maneuver the motorcycle in a
satisfactory manner. At one level, increasing the knowledge
conatined in the KB generally produces an increase in the size of
the KB. A large KB is difficult to store and requires relatively
large amounts of computational reasources in the soft computing
controller. What is missing from the prior art is a system and
method for controlling a motorcycle using a reduced-size KB that
provides sufficient knowledge to provide good control.
SUMMARY
The present invention solves these and other problems by providing
a SC optimizer for designing a globally-optimized KB to be used in
a SC system for an electronically-controlled motorcycle. In one
embodiment, the SC optimizer includes a fuzzy inference engine. In
one embodiment, the fuzzy inference engine includes a Fuzzy Neural
Network (FNN). In one embodiment, the SC Optimizer provides Fuzzy
Inference System (FIS) structure selection, FIS structure
optimization method selection, and teaching signal selection.
In one embodiment, the user makes the selection of fuzzy model,
including one or more of: the number of input and/or output
variables; the type of fuzzy inference model (e.g., Mamdani,
Sugeno, Tsukamoto, etc.); and the preliminary type of membership
functions.
In one embodiment, a Genetic Algorithm (GA) is used to optimize
linguistic variable parameters and the input-output training
patterns. In one embodiment, a GA is used to optimize the rule
base, using the fuzzy model, optimal linguistic variable
parameters, and a teaching signal.
One embodiment, includes optimization of the FIS structure by using
a GA with a fitness function based on a response of a model of
motorcycle and rider system.
One embodiment, includes optimization of the FIS structure by a GA
with a fitness function based on a response of the actual
motorcycle/rider system.
The result is a specification of an FIS structure that specifies
parameters of the optimal FC according to desired requirements.
BRIEF DESCRIPTION OF THE FIGURES
The above and other aspects, features, and advantages of the
present invention will be more apparent from the following
description thereof presented in connection with the following
drawings.
FIG. 1 shows a robust intelligent control system of motorcycle and
rider navigation model.
FIG. 2A is a photograph of an electric scooter used for
demonstration purposes.
FIG. 2B shows a computer model of the scooter from FIG. 2A.
FIG. 3 shows a simulation model of the scooter from FIG. 2B.
FIG. 4 shows a model of a spring and damper system.
FIG. 5 shows a model for aerodynamic forces (drag and lift).
FIG. 6 shows a tire model.
FIG. 7 shows a tire force coordinate system (Xt,Yt,Zt).
FIGS. 8A-B show tire modeling using a JARI tire, including tire
data (A), and tire model (B).
FIG. 9 shows an eigenvector of weave mode.
FIG. 10 shows a locus of eigenvalues of the weave and wobble mode
for various velocities.
FIG. 11 shows features of the SC Optimizer.
FIG. 12A is a flowchart of the SC Optimizer.
FIG. 12B is a flowchart for using SC Optimizer.
FIG. 12C shows a flowchart for evaluating the SC Optimizer.
FIG. 13 shows a structure of the fuzzy control system and
development of the fuzzy controller.
FIG. 14 shows a structure of the feedforward controller and its
development.
FIGS. 15A-B show geometrical parameters of a rider model and
course, including reference roll angle (A), and length of reference
and deviation (B).
FIG. 16 shows the structure of a simulation to gain the teaching
signals used by the SC optimizer.
FIGS. 17A-G show examples of the data during fitness function
evaluation simulation.
FIGS. 18A-C show example results of the optimized parameters for
different initial conditions, including Kp1 (A), Kp2 (B), and Kd2
(C).
FIG. 19 shows membership functions for the fuzzy controller.
FIG. 20 shows outputs of the fuzzy controller.
FIG. 21 shows a simulation model for the fuzzy controller.
FIGS. 22A-B show the map of the actual test course and simulated
course.
FIGS. 23A-F show the results of the simulation with the fuzzy
controller (without excitation).
FIGS. 24A-E show the results of the simulation with fuzzy
controller (with excitation).
FIGS. 25A-D show simulation results with different transport
delay.
FIG. 26A-B show the look-forward model and the process to generate
a feedforward rider model, where (A) shows the relation between
deviation of course at length of reference and turning radius, and
(B) shows the process of assembling the g feedforward rider
model.
FIG. 27 shows a model for circular course simulation.
FIG. 28A-F show the optimized parameters.
FIG. 29A-B shows the relation between computed reference roll angle
and the optimized parameter.
FIG. 30A-B show course lane change and model for lane change
simulation.
FIG. 31 shows the feedforward model.
FIG. 32A-E show results of the simulation with a hold torque
feedforward controller.
FIG. 33A-D show results of the simulation with transport delay.
FIG. 34 shows the structure of the measurement system.
FIG. 35 shows a test scooter with telemetry system.
FIG. 36A-B show an example of monitoring display (A) and measured
data (B).
DETAILED DESCRIPTION
FIG. 1 is a block diagram of an intelligent motorcycle (with rider)
navigation control system based on soft computing and using a soft
computing optimizer to generate a Knowledge Base (KB) 108 for a
Fuzzy Controller (FC) 109. The motorcycle and rider system is
represented by a block 102. Output from the block 102 is provided
to a fitness function in block 104 of a GA in block 105. The
objective information for the knowledge base (KB) 108 is extracted
(from the output of block 102) by a Genetic Analyzer (GA) 105 using
the results of stochastic simulation of the motorcycle/rider
dynamic behavior. The GA 105 and Fitness Function 104 are
components of a System Simulation of Control Quality (SSCQ)
242.
Using a set of inputs, and a fitness function in block 104, the GA
in block 105 works in a manner similar to an evolutionary process
to arrive at a solution, which is, hopefully, optimal. The GA in
block 105 generates sets of "chromosomes" (that is, possible
solutions) and then sorts the chromosomes by evaluating each
solution using the fitness function in block 104. The fitness
function in block 104 determines where each solution ranks on a
fitness scale. Chromosomes (solutions), which are more fit, are
those, which correspond to solutions that rate high on the fitness
scale. Chromosomes, which are less fit, are those, which correspond
to solutions that rate low on the fitness scale.
Chromosomes that are more fit are kept (survive) and chromosomes
that are less fit are discarded (die). New chromosomes are created
to replace the discarded chromosomes. The new chromosomes are
created by crossing pieces of existing chromosomes and by
introducing mutations.
The motorcycle/rider model 102 is controlled by a linear P(D)
controller 101. In one embodiment, the controller 100 uses both a
PD controller 1012 and a P controller 1011 to control different
aspects of the motorcycle system. The P(D) controller 101 has a
linear transfer function and thus is based upon a linearized
equation of motion for the control object in block 102. Prior art
GAs used to program P(D) controllers typically use simple fitness
function and thus do not solve the problem of poor controllability
typically seen in linearization models. As is the case with most
optimizers, the success or failure of the optimization often
ultimately depends on the selection of the performance (fitness)
function.
Evaluating the motion characteristics of a nonlinear plant is often
difficult, in part due to the lack of a general analysis method.
Conventionally, when controlling a plant with nonlinear motion
characteristics, it is common to find certain equilibrium points of
the plant and the motion characteristics of the plant are
linearized in a vicinity near an equilibrium point. Control is then
based on evaluating the pseudo (linearized) motion characteristics
near the equilibrium point.
Computation of optimal control based on soft computing includes the
GA in block 105 as the first step of global search for optimal
solution on a fixed space of solutions. The GA searches for a set
of control weights for the plant. Firstly the weight vector
K={k.sub.1, . . . , k.sub.n} is used by a conventional P(D)
controller in block 101 in the generation of a signal .delta.(K)
which is applied to the plant. The entropy S(.delta.(K)) associated
with the behavior of the plant on this signal is assumed as a
fitness function to be minimized. The GA is repeated several times
with regular time intervals in order to produce a set of weight
vectors. The vectors generated by the GA in block 105 are then
provided to a FNN and the output of the FNN to a fuzzy controller
in block 109. The output of the fuzzy controller in block 109 is a
collection of gain schedules for the linear controllers in block
101 that control the motorcycle 102.
So in summary, designing the intelligent control system includes
two stages. Stage 1 involves finding a teaching signal (also
referred to as teaching patterns). Stage 2 involves approximation
of optimal control.
Finding teaching patterns (input-output pairs) of optimal control
by using the GA are aspects of the SSCQ 242, based on the
mathematical model of the controlled motorcycle and a physical
criteria of minimum of entropy production rate.
Approximation of the optimal control (from Stage 1) by the
corresponding Fuzzy Controller (FC). This FC is referred to as the
optimal Fuzzy Controller.
The first stage is the acquisition of a robust teaching signal of
optimal control without the loss of information. The output of
first stage is the robust teaching signal, which contains
information about the controlled object behavior and corresponding
behavior of the control system.
The second stage is the approximation of the teaching signal by
building of a fuzzy inference system. The output of the second
stage is the knowledge base (KB) 108 for the FC 109.
The design of the optimal KB 108 includes specifying the optimal
numbers of input-output membership functions, their optimal shapes
and parameters, and a set of optimal fuzzy rules.
In one embodiment for the Stage 2 realization, the optimal FC 109
is obtained using a FNN with the learning method based on the error
back-propagation algorithm. The error back-propagation algorithm is
based on the application of the gradient descent method to the
structure of the FNN. The error is calculated as a difference
between the desired output of the FNN and an actual output of the
FNN. Then the error is "back propagated" through the layers of the
FNN, and parameters of each neuron of each layer are modified
towards the direction of the minimum of the propagated error.
However, back-propagation algorithm has a few disadvantages. In
order to apply back-propagation approach it is necessary to know
the complete structure of the FNN prior to the training
optimization. The back-propagation algorithm cannot be applied to a
network with an unknown number of layers or an unknown number
nodes. The back-propagation process cannot automatically modify the
types of the membership functions. Usually, the initial state of
the coefficients for back-propagation algorithm is set up randomly,
and as a result, the back-propagation algorithm often finds only a
"local" optimum close to the initial state.
In spite of the above disadvantages, the error back-propagation
algorithm is commonly used in practice. For example, the Adaptive
Fuzzy Modeler (AFM) provided by STMicroelectronics Inc. creates
Sugeno 0 order fuzzy inference systems using the error
back-propagation algorithm.
The algorithm used in the AFM is based on the following two steps.
In the first step, a user specifies the parameters of a future FNN
such as the numbers of inputs and outputs and the number of fuzzy
sets for each input/output. The AFM "optimizes" the rule base,
using a so-called "let the best rule win" (LBRW) technique. During
this phase, the membership functions are fixed as uniformly
distributed among the universe of discourse, and AFM calculates the
firing strength of the each rule, eliminating the rules with zero
firing strength, and adjusting centers of the consequents of the
rules with nonzero firing strength. It is possible during
optimization of the rule base specify the learning rate parameter,
depending on the current problem. In AFM there is also an option to
build the rule base manually. In this case, user can specify the
centroids of the input fuzzy sets, and then according to the
specification, the system builds the rule base automatically.
In the second step, the AFM builds the membership functions. The
user can specify the shape factors of the input membership
functions. Shape factors supported by AFM are: Gaussian, Isosceles
Triangular, and Scalene Triangular. The user must also specify the
type of fuzzy operation in Sugeno model: supported methods are
Product and Minimum.
After specifying the membership function shape and Sugeno inference
method, the AFM starts optimization of the membership function
shapes, using the structure of the rules, developed during Step 1.
There are also some optional parameters to control optimization
rate such as a target error and the number of iterations, the
network should make. The termination condition on the optimization
is reaching of the number of iterations, or when the error reaches
its target value.
AFM inherits the weakness of the back-propagation algorithm
described above, and the same limitations. The user must specify
types of membership functions, number of membership functions for
each linguistic variable and so on. The rule number optimizer in
the AFM is called before membership functions optimization, and as
a result the system often becomes unstable during membership
function optimization phase.
However, some of the weakness of system such as the AFM can be
ameliorated by optimizing the structure of the KB before training
the KB.
In FIG. 1, the motorcycle (MC) and rider model (control algorithm)
are shown as a simulation model 102. FIG. 2A shows an example of a
real MC (in this cas, an electric scooter) and FIG. 2B shows a
model image of the scooter from FIG. 2A.
FIG. 3 shows one embodiment of a simulation motorcycle/rider model
developed by using SimMechanics (a MATLAB add-on). Various rider
models, including, for example, a fuzzy controller model and a hold
torque feedforward model, can be used.
In one embodiment, the models describe the stability of the MC
during a straight line run. The model descriptions of the MC in
this approache is provided by conventional straight-line modeling
techniques. Such straight-line models typically do not include a
steering model but do typically include a passive rider model
(e.g., a hands on and hands off model).
In one embodiment, the model is similar to a model constructed
using the propular DADS modeling program from LMS International. In
this model, the forward velocity is fixed. The rider steering
models structure is similar to the fuzzy controller model, but the
parameters of the rider model use fixed values which can be
selected by trial and error.
In one embodiment, the MC model is constructed using DADS, with a
rider steering model based on a look-forward model. The parameters
are fixed for each course. Lane change simulation and sine course
simulation are shown.
In one embodiment, the control behaviour of motorcycle and rider is
described. The motorcycle model of the simulation is substantially
simialr to the model described above. The rider model is a
look-forward model. The parameters were fixed for each course. Lane
change simulation is shown.
In one embodiment, various driving maneuvers such as cornering and
double lane change with a driver control algorithm are used with a
motorcycle model using the ADAMS program available from MSC
Software Corporation. The driver control model has two parts: a
velocity control model and a steering control model. In this
steering control model, the desired camber angle is estimated and
depends on the driving maneuver. The parameters for the controller
can be fixed.
In one embodiment, the stability control of a motorcycle/rider
system based on fuzzy control in conjunction with a GA and an
auto-tuning method is used. The model used in simulation need not
be a true motorcycle model. For example, an inverted pendulum
hinged to a rotating disk can be used as a representative model of
the motorcycle.
One embodiment provides an autonomous two-wheeled vehicle and
simulation results. The rider model can be divided into two parts:
standing control and directional control. The control algorithm of
standing stability is based on feedback control with gain
parameters that depend on velocity. Directional control is the
target roll angle, which can be determined by predicted course
error. In one embodiment a spring-damper is connected between the
steering bar and the actuator for steering to stabilize the
motorcycle (which makes the control method relatively more
complex).
The effect on the perception of riding comfort issue for changing
mass center position, total weight on the MC, the wheelbase, the
front fork rake angle and the front wheel trail distance of the MC
is discussed. The tire model is described in connection with FIGS.
6-8.
FIG. 3 shows a simulation model of the scooter shown in FIGS. 2A-B.
The model in the FIG. 3 was made using SimMechanics which is an
optional tool for kinematics analysis using the popular MATLAB
computer program.
Table 1 shows the model notations and parameter definitions.
TABLE-US-00001 TABLE 1 Used notations Model notations .tau..sub.d:
Drive Torq(Nm) .tau..sub.z: Steer Torq(Nm) .tau..sub.hold: steer
torq to hold roll angle(Nm) .tau..sub.1: lower body controlling
Torq(Nm) .tau..sub.2: upper body controlling Torq(Nm) K.sub.p1,
K.sub.p2, K.sub.d2,,,,: gains .nu.: forward velocity(m/s)
.nu..sub.ref: reference velocity(m/s) .phi.: roll angle(rad)
.phi..sub.ref: reference roll angle(rad) .phi..sub.ref0: calculated
reference roll angle(rad) .phi..sub.col: correction for reference
roll angle(rad) .phi.: rollrate(rad/s) .alpha.: required lateral
acceleration(m/s.sup.2) g: gravity acceleration(m/s.sup.2) y:
deviation at LR(m) y.sub.0: deviation (course err)(m) t: time(sec)
LR: length of reference(m) FF: fitness function .omega.:
yawrate(rad/s) R: turning radius(m) .theta.: tire yaw angle in
h@rizontal plane(FIG. 7) .gamma.: camber angle(rad)(FIG. 7)
.alpha.: lateral slip(rad) .kappa.: rolling slip h: height of tire
center(m)(FIG. 7) tr: Tire original radius(m) te: Tire effective
radius(m)(FIG. 7) .omega.s: tire spin velocity(rad/s) (x.sub.r,
y.sub.r, z.sub.r): directional vector of the tire rotational axis
in gloval coordinate(m) (vx, vy, vz): velocity vector of tire
center point in gloval coordinate(m/s) (vx.sub.1, vy.sub.1,
vz.sub.1): velocity vector of tire center point in tire force
coordinate(m/s)(FIG. 7) (F.sub.X.alpha., F.sub.Y.alpha.,
F.sub.Z.alpha.): force vector of tire at contact point in tire
force coordinate(N)(FIG. 7) (T.sub.X.alpha., 0, T.sub.Z.alpha.):
moment vector of tire at contact point in tire force coordinate(Nm)
F.sub.Y.alpha.0: static lateral force of tire at contact point in
tire force coordinate(N) T.sub.X.alpha.0: static overturning moment
of tire at contact point in tire force coordinate(Nm)
T.sub.Z.alpha.0: static aligning torque of tire at contact point in
tire force coordinate(Nm) T.sub.X1: moment of tire excepting
rolling moment around Xt(FIG. 7) direction in tire force coordinate
at center point of tire (Nm) T.sub.Y1: moment of tire excepting
rolling moment around Yt(FIG. 7) direction in tire force coordinate
at center point of tire (Nm) (F.sub.X, F.sub.Y, F.sub.Z): force
vector of tire at center point of tire in global coordinate(N)
(T.sub.X, T.sub.Y, T.sub.Z): moment vector of tire excepting
rolling moment at center point of tire in global coordinate(Nm)
M.sub.roll: rolling moment of tire(Nm) K: Tire vertical spring
constant(N/m) D: Tire vertical damping(Ns/m) C.sub.1: traction
coeficient .nu..sub.1: forward speed for tire(m/s) .sigma.:
coeficient for relaxation length(m) .sigma..sub.1: coef. for
relaxation length proportion to load .sigma..sub.2: coef. for
relaxation length constant Cy.sub.1, Cy.sub.11, Cy.sub.12,
Cy.sub.3, Cy.sub.31, Cy.sub.32: coef. lateral force Cz.sub.1,
Cz.sub.11, Cz.sub.12, Cz.sub.3, Cz.sub.31, Cz.sub.32: coef.
overturning moment Cx.sub.1, Cx.sub.11, Cx.sub.12, Cx.sub.3,
Cx.sub.31, Cx.sub.32: coef. aligning torque d.sub.Fy, d.sub.Fz,
d.sub.Tz, d.sub.Tx: noises
The MC plant model includes: 7 rigid bodies (mainframe in block
3001, rider in block 3002, front fork in block 3003, front arm in
block 3004, rear arm in block 3005, front & rear wheels in
block 3006 and in block 3007, respectively); 5 rotating joints
(rider roll in block 3008, steer in block 3009, rear arm pivot in
block 3010, front & rear wheels in block 3011 and in block
3012, respectively); 1 slide joint (front suspension in block
3013); 1 combination joint in block 3014 (a 6 degree of freedom
main body to inertial system); 4 spring & damper (front
suspension in block 3015, steer in block 3016, rider in block 3017,
rear suspension in block 3018); an external force in block 3019
(air force(drag & lift); and a tire force in block 3020 for the
front tire and a tire forces in block 3021 for the rear tire.
FIG. 4 shows the spring and damper model used in blocks 3015 and
3018. Parameters of the nonlinear spring, represented as look-up
table, are provided to the block 4001. Damping coefficient are
provided to the block 4002.
FIG. 5 shows the aerodynamic force model used in block 3019. Drag
and lift parameters are provided to block 5001 and to block 5002,
respectively.
In the present disclosure, results from a MC simulation model are
compared with results from an actual demonstration model. The
differences between the MC model developed and the demonstration
model are: 1. The linear tire force model in block 3020 and in
block 3021 with first order lag was applied in the MC model
(instead of the complex model used in the demonstration). 2. Noise
road signals were added to the tire model. 3. Coordinate system was
changed: +Z points up in the MC model and +Z points down in the
demonstration. 4. Front and rear brakes were removed in the MC
model. 5. Twist freedom of steering shaft was removed and only
steering rotation freedom remains in block 3009. The removed twist
axis was perpendicular axis to the steering shaft in central plane.
The reason to remove the freedom is that in this model, linear tire
model was used and small elastic deformation should be ignored in
view of balance of accuracy.
As mentioned above, blocks 3020 and 3021 represented the linear
tire model. Tire properties are an important determinant of MC
maneuverability and stability, and tire models are thus important
to behavior simulations employing a mechanical analysis. FIG. 6 is
a block diagram of the tire model, where tire axis sensor data 6003
and tire spin sensor data are provided to a tire model 6001. The
tire model 6001 provides rolling moment data 6005 and force moment
and tire axis data 6006. The tire mathematical model in block 6001
is designed using the following equations: .theta.=-arctan
(x.sub.r/y.sub.r) (5-1) .gamma.=arcsin (z.sub.r/ {square root over
((x.sub.r.sup.2+y.sub.r.sup.2+z.sub.r.sup.2))}) (5-2) vx.sub.t=vx
cos .theta.+vy sin .theta. (5-3) vy.sub.t=-vx sin .theta.+vy cos
.theta. (5-4) vz.sub.t=vz (5-5) te=h/cos .gamma. (5-6)
.kappa.=-(vx.sub.t-te.omega.s)/vx.sub.t (5-7)
.alpha.=vy.sub.t/vx.sub.t (5-8)
F.sub.Zct=(K(tr-te)-Dte)(1+d.sub.Fz) (5-9)
F.sub.Xct=C.sub.t.kappa.F.sub.Zct (5-10)
F.sub.Yct0=(Cy.sub.1.alpha.+Cy.sub.3.gamma.)(1+d.sub.Fy) (5-11)
T.sub.Zct0=(Cz.sub.1.alpha.+Cz.sub.3.gamma.)(1+d.sub.Tz) (5-12)
T.sub.Xct0=Cx.sub.1.alpha.+Cx.sub.3.gamma.(1+d.sub.Tx) (5-13)
Cy.sub.1=Cy.sub.11F.sub.Zct0+Cy.sub.12 (5-14)
Cy.sub.3=Cy.sub.31F.sub.Zct0+Cy.sub.32 (5-15)
Cz.sub.1=Cz.sub.11F.sub.Zct0+Cx.sub.12 (5-16)
Cz.sub.3=Cz.sub.31F.sub.Zct0+Cz.sub.32 (5-17)
Cx.sub.1=Cx.sub.11F.sub.Zct0+Cx.sub.12 (5-18)
Cx.sub.3=Cx.sub.31F.sub.Zct0+Cx.sub.32 (5-19)
.sigma.=.sigma..sub.1F.sub.Zct+.sigma..sub.2 (5-20)
.sigma..times..times..times..times..times..sigma..times..times..times..ti-
mes..times..sigma..times..times..times..times..times. ##EQU00001##
M.sub.roll=T.sub.Zct sin .gamma.-F.sub.Xctte (5-24)
F.sub.X=F.sub.Xct cos .theta.-F.sub.Yct sin .theta. (5-25)
F.sub.Y=F.sub.Xct sin .theta.+F.sub.Yct cos .theta. (5-26)
F.sub.Z=F.sub.Zct (5-27) T.sub.Xt=T.sub.Xct+F.sub.Yctte cos
.gamma.+F.sub.Zctte sin .gamma. (5-28) T.sub.yt=-T.sub.Zct cos
.gamma. sin .gamma. (5-29) T.sub.t=T.sub.Xt cos .theta.-T.sub.Yt
sin .theta. (5-30) T.sub.Y=T.sub.Xt sin .theta.+T.sub.Yt cos
.theta. (5-31) T.sub.Z=T.sub.Zct cos.sup.2 .gamma. (5-32)
Inputs for the tire model in block 6001 are tire spin velocity
in-block 6002 and the data of tire axis in block 6003. The data of
the tire axis in block 6003 contain the height h of tire center and
vector of tire axis (x.sub.r,y.sub.r,z.sub.r) and velocity vector
of tire center (v.sub.x,v.sub.y,v.sub.z). From these data, tire yaw
angle, camber angle, velocity vector of tire center in tire force
coordinate, tire effective radius, rolling slip, and lateral slip
are calculated by Eqs (5-1)-(5-8) (see FIG. 7 for details).
Static tire forces and moments at contact point in tire force
coordinate are calculated by Eqs (5-9)-(5-20). Noise signals in
block 6004 can be added to the tire model during these
calculations. Output data from the tire model 6001 include:
vertical force, driving force, static side force, static self
aligning torque, static overturning torque at the contact point,
and relaxation length. Conversion from static force and moments to
dynamic moments is provided by Eqs (5-21).about.(5-23).
The rolling moment in block 6005 for tire can be split by Eq.
(5-24). The conversion from forces and moments at contact point to
that at the shaft of tires in global coordinate system in block
6006 is provided by Eqs (5-25)-(5-32).
The tire data for a desired tire can be measured or obtained, for
example, from organizations such as the Japan Automobile Research
Institute (JARI). In one embodiment, tires of the same
specification can be used for the front and rear.
In one embodiment, input data from a JARI tire model are vertical
force, camber angle, and slip angle. Output data are side force,
self aligning torque, overturning torque, and relaxation length.
FIG. 8A shows a sample tire data file containing data provided by
JARI. This tire model is a linear model and thus less accurate than
non-linear models for large camber angles and large slip angles.
Nevertheless, the linear model is adequate for many circumstances,
including, but not limited to, modeling the scooter of FIG. 2A in
typical operating regimes.
FIG. 8B shows a transformation of the JARI tire model to add
caculation of transient effects to the tire model. In FIG. 8B input
blocks, provide for arranging input data from sensed data of
posistion, velocity, and posture of the tire. Additoinal blocks are
provided to convert output data at tire contact in tire coordinate
system to force and moment at tire center in inertia coordinate
system containing transient effect. Forward and backward force
calculation are provided to evaluate drag and driving forces.
The model of the MC in SimMechanics is a complicated numerical
simulation model that predicts the operation of the motorcycle
system. In one embodiment, Eigenvalue analysis is used in
connection with the numerical model to extract and explore various
characteristics of the MC model produced by the simulation.
For eigenvalue analysis, linearization of the model in the straight
balanced running model can be used. The model has 12 degrees of
freedom and adding differentiation to those 12 degrees, gives are
24 degrees of freedom. Moreover, there are six degrees of freedom
that originated in the feedback model of the relaxation length used
in order to take the dynamic character of a tire into
consideration. As a result of linearization of the model, a
30.times.30 matrix representation of the system is obtained,
correspdingly, 30 eigenvalues and 30 eigenvectors are calculated.
The characteristics of the motion of the model are determined by a
relatively few modes among the 30 modes. The MC weave modes and
wobble modes can be unstable, and the corredponding eigenvalues and
eigenvectors are relatively large.
FIG. 9 shows the eigenvector of weave mode corresponding to 7
m/sec. The mode is a coupling mode that includes cyclic motions of
sway, roll, yaw and swing of steer. Yaw rate and swing of steer are
in phase and roll angle is out of phase.
FIG. 10 shows the locus of eigenvalues of weave and wobble modes
for various velocities. The weave mode is unstable for velocities
under 10 m/sec. Wobble 1 mode corresponds to a simple swing of the
steering. When the damping coefficient of steering is small, the
mode will be unstable. In this model the damping coefficient
C.sub.st=5 is adopted. Wobble 2 mode includes rider roll, chassis
roll, and steering swing. Wobble 3 mode includes yaw, roll, and
steering swing. Table 2 shows eigenvalues of weave and wobble 1
when the damping coefficient of steering changes.
TABLE-US-00002 TABLE 2 Coefficient C.sub.st Weave eigenvalue Wobble
1 eigenvalues 0 0.55 + 3.3i 12 + 57i 1 0.63 + 3.3i 8.0 + 58i 3 0.76
+ 3.2i 1.2 + 60i 5 0.88 + 3.0i -6.7 + 61i
The results of simulation by the rider models were both stable.
Robustness of the controllers was confirmed by simulation taking
into account the transmission delay of the response of actuator and
noises in tire forces (as described below).
In one embodiment, Stage 2 (defining the structure of the KB) is
realized using a new structure based on Soft Computing (SC)
techniques. The structure of the intelligent control system is
based on the SC optimizer shown in FIG. 1. In this case, the SC
optimizer is used in place of the FNN block.
The SC optimizer generates a KB of Fuzzy Inference System (FIS)
from the digital in-out data. The SC optimizer provides GA based
FNN learning, including rule extraction and KB optimization. In one
embodiment, the SC optimizer can use as a teaching signal both
pattern file and the model (control object) output.
FIG. 11 shows the features of the SC optimizer. The SC optimizer
includes a fuzzy inference system (FIS) represented in the form of
a FNN. In one embodiment, the SC optimizer allows structure
selection, such as for example, Sugeno FIS order 0 and 1, Mamdani
FIS, Tsukamoto FIS, etc. In one embodiment, the SC optimizer allows
structure optimization selection, such as for example, Optimization
via Genetic Algorithm (GA), Linguistic variables optimization, and
rule-based optimization. In one embodiment, the SC optimizer allows
teaching source selection, such as for example: teaching signal as
a look-up table (e.g., a table of in-out patterns); and/or teaching
signal as a fitness function. When provided by a fitness function,
the teaching signal fitness function can be: calculated as a
dynamic system response (e.g., using in MATLAB), calculated as a
result of control of real control object (e.g., obtained using
dSPACE or other method of connection of control object directly to
computer), etc.
FIG. 12 is a flowchart for one embodiment of the SC optimizer. By
way of explanation, and not by way of limitation, the operation of
the flowchart divides operation in to five stages, shown as Stages
1, 2, 3, 4, and 5.
In Stage 1, the user selects a fuzzy model by selection one of
parameters such as, for example, the number of input and output
variables, the type of fuzzy inference model (Mamdani, Sugeno,
Tsukamoto, etc.), source of the teaching signal.
In Stage 2, a first GA (GA1) optimizes linguistic variable
parameters, using the information obtained in Stage 1 about the
general system configuration, and the input-output training
patterns, obtained from the training signal as an input-output
tables.
In Stage 3 a precedent part of the rule base is created and rules
are ranked according to their firing strength. Rules with high
firing strength are kept, whereas weak rules with small firing
strength are eliminated.
In Stage 4, a second GA (GA2) optimizes a rule base, using the
fuzzy model obtained in Stage 1, optimal linguistic variable
parameters obtained in Stage 2, selected set of rules obtained in
Stage 3 and the teaching signal.
In Stage 5, the structure of FNN is further optimized. In order to
reach the optimal structure, the classical derivative-based
optimization procedures can be used, with a combination of initial
conditions for back propagation, obtained from previous
optimization stages. The result of Stage 5 is a specification of
fuzzy inference structure that is optimal for the plant 120. Stage
5 is optional and can be bypassed. If Stage 5 is bypassed, then the
FIS structure obtained with the GAs of Stages 2 and 4 is used.
In one embodiment the teaching signal, representing one or more
input signals and one or more output signals, is divided into input
and output parts. Each of the parts is divided into one or more
signals. Thus, in each time point of the teaching signal there is a
correspondence between the input and output parts.
Each component of the teaching signal (input or output) is assigned
to a corresponding linguistic variable, in order to explain the
signal characteristics using linguistic terms. Each linguistic
variable is described by some unknown number of membership
functions, like "Large", "Medium", "Small", etc.
"Vertical relations" represent the explicitness of the linguistic
representation of the concrete signal, e.g. how the membership
functions is related to the concrete linguistic variable.
Increasing the number of vertical relations will increase the
number of membership functions, and as a result will increase the
correspondence between possible states of the original signal, and
its linguistic representation. An infinite number of vertical
relations would provide an exact correspondence between signal and
its linguistic representation, because to each possible value of
the signal would be assigned a membership function, but in this
case the situations as "over learning" may occur. Smaller number of
vertical relations will increase the robustness, since some small
variations of the signal will not affect much the linguistic
representation. The balance between robustness and precision is a
very important moment in design of the intelligent systems, and
usually this task is solved by Human expert.
"Horizontal relations" represent the relationships between
different linguistic variables. Selected horizontal relations can
be used to form components of the linguistic rules.
To define the "horizontal" and "vertical" relations mathematically,
consider a teaching signal: [x(t),y(t)], Where: t=1, . . . ,
N--time stamps; N--number of samples in the teaching signal;
x(t)=(x.sub.1(t), . . . x.sub.m(t))--input components;
y(t)=(y.sub.1(t), . . . y.sub.n(t))--output components.
Define the linguistic variables for each of the components. A
linguistic variable is usually defined as a quintuple:
(x,T(x),U,G,M), where x is the name of the variable, T(x) is a term
set of the x, that is the set of the names of the linguistic values
of x, with a fuzzy set defined in U as a value, G is a syntax rule
for the generation of the names of the values of the x and M is a
semantic rule for the association of each value with its meaning.
In the present case, x is associated with the signal name from x or
y, term set T(x) is defined using vertical relations, U is a signal
range. In some cases one can use normalized teaching signals, then
the range of U is [0,1]. The syntax rule G in the linguistic
variable optimization can be omitted, and replaced by indexing of
the corresponding variables and their fuzzy sets.
Semantic rule M varies depending on the structure of the FIS, and
on the choice of the fuzzy model. For the representation of all
signals in the system, it is necessary to define m+n linguistic
variables:
Let [X,Y], X=(X.sub.1, . . . , X.sub.m), Y=(Y.sub.1, . . . ,
Y.sub.n) be the set of the linguistic variables associated with the
input and output signals correspondingly. Then for each linguistic
variable one can define a certain number of fuzzy sets to represent
the variable:
.times..times..mu..times..mu..times..times..times..mu..times..mu.
##EQU00002##
.times..times..mu..times..mu..times..times..times..mu..times..mu.
##EQU00002.2##
Where .mu..sub.X.sub.i.sup.j.sup.i, i=1, . . . , m, j.sub.i=1, . .
. , l.sub.X.sub.i are membership functions of the i th component of
the input variable; and .mu..sub.Y.sub.i.sup.j.sup.i, i=1, . . . ,
j.sub.i=1, . . . , l.sub.Y.sub.i are membership functions of the i
th component of the output variable.
Usually, at this stage of the definition of the KB, the parameters
of the fuzzy sets are unknown, and it may be difficult to judge how
many membership functions are necessary to describe a signal. In
this case, the number of membership functions (power of term set),
l.sub.X.sub.i.di-elect cons.[1, L.sub.MAX], i=1, . . . , m can be
considered as one of the parameters for the GA (GA1) search, where
L.sub.MAX is the maximum number of membership functions allowed. In
one embodiment, L.sub.MAX is specified by the user prior to the
optimization, based on considerations such as the computational
capacity of the available hardware system.
Knowing the number of membership functions, it is possible to
introduce a constraint on the possibility of activation of each
fuzzy set, denoted as p.sub.X.sub.i.sup.j.
One of the possible constraints can be introduced as:
.times..times.'.gtoreq..times..times. ##EQU00003##
This constraint will cluster the signal into the regions with equal
probability, which is equal to division of the signal's histogram
into curvilinear trapezoids of the same surface area. Supports of
the fuzzy sets in this case are equal or greater to the base of the
corresponding trapezoid. How much greater the support of the fuzzy
set should be, can be defined from an overlap parameter. For
example, the overlap parameter takes zero, when there is no overlap
between two attached trapezoids. If it is greater than zero then
there is some overlap. The areas with higher probability will have
in this case "sharper" membership functions. Thus, the overlap
parameter is another candidate for the GA1 search. The fuzzy sets
obtained in this case will have uniform possibility of
activation.
Modal values of the fuzzy sets can be selected as points of the
highest possibility, if the membership function has unsymmetrical
shape, and as a middle of the corresponding trapezoid base in the
case of symmetric shape. Thus one can set the type of the
membership functions for each signal as a third parameter for the
GA1.
The relation between the possibility of the fuzzy set and its
membership function shape can also be found. The possibility of
activation of each membership function is calculated as
follows:
.function..mu..times..times..mu..function..function.
##EQU00004##
Mutual possibility of activation of different membership functions
can be defined from a geometrical view point as:
.function..times..mu..mu..times..times..mu..function..function..mu..funct-
ion..function. ##EQU00005## where * denotes selected T-norm (Fuzzy
AND) operation; j=1, . . . , l.sub.X.sub.i, l=1, . . . ,
l.sub.X.sub.k, are indexes of the corresponding membership
functions.
In fuzzy logic literature, T-norm, denoted as * is a two-place
function from [0,1].times.[0,1] to [0,1]. It represents a fuzzy
intersection operation and can be interpreted as minimum operation,
or algebraic product, or bounded product or drastic product.
S-conorm, denoted by {dot over (+)}, is a two-place function, from
[0,1].times.[0,1] to [0,1]. It represents a fuzzy union operation
and can be interpreted as algebraic sum, or bounded sum and drastic
sum. Typical T-norm and S-conorm operators are presented in Table
3.
TABLE-US-00003 TABLE 3 T-norms (fuzzy intersection) S-conorms
(fuzzy union) min(x, y) - minimum operation max(x, y) - maximum
operation xy - algebraic product x + y - xy - algebraic sum x * y =
max[0, x + y - 1] -bounded product
.times..times..function..times..times. ##EQU00006##
.times..times..times..times..times..times..times..times.<
##EQU00007##
.times..times..times..times..times..times..times..times..times..times.&g-
t; ##EQU00008##
If i=k, and j.noteq.1, then equation (6.2) defines "vertical
relations"; and if i.noteq.k, then equation (6.2) defines
"horizontal relations". The measure of the "vertical" and of the
"horizontal" relations is a mutual possibility of the occurrence of
the membership functions, connected to the correspondent
relation.
The set of the linguistic variables is considered as optimal, when
the total measure of "horizontal relations" is maximized, subject
to the minimum of the "vertical relations".
Hence, one can define a fitness function for the GA1 which will
optimize the number and shape of membership functions as a maximum
of the quantity, defined by equation (6.2), with minimum of the
quantity, defined by equation (6.1).
The chromosomes of the GA1 for optimization of linguistic variables
according to Equations (6.1) and (6.2) have the following
structure:
.times. .times..alpha..times..alpha. .times..times. ##EQU00009##
Where:
l.sub.X(Y).sub.i.di-elect cons.[1, L.sub.MAX] are genes that code
the number of membership functions for each linguistic variable
X.sub.i(Y.sub.i);
.alpha..sub.X(Y).sub.i are genes that code the overlap intervals
between the membership functions of the corresponding linguistic
variable X.sub.i(Y.sub.i); and
T.sub.X(Y).sub.i are genes that code the types of the membership
functions for the corresponding linguistic variables.
Another approach to the fitness function calculation is based on
the Shannon information entropy. In this case instead of the
equations (6.1) and (6.2), for the fitness function representation
one can use the following information quantity taken from the
analogy with information theory:
.times..times..function..times..function..mu..times..function..function..-
mu..times..times..times..mu..function..function..times..function..mu..func-
tion..function..times..times..times..times..function..times..mu..mu..times-
..times..times..mu..function..function..mu..function..function..times..fun-
ction..mu..function..function..mu..function..function..times.
##EQU00010##
In this case, GA1 will maximize the quantity of mutual information
(6.2a), subject to the minimum of the information about each signal
(6.1a). In one embodiment the combination of information and
probabilistic approach can also be used.
In case of the optimization of number and shapes of membership
functions in Sugeno--type FIS, it is enough to include into GA
chromosomes only the input linguistic variables. The detailed
fitness functions for the different types of fuzzy models will be
presented in the following sections, since it is more related with
the optimization of the structure of the rules.
In one embodiment, a rules pre-selection algorithm selects the
number of optimal rules and their premise structure prior
optimization of the consequent part. Consider the structure of the
first fuzzy rule of the rule base R.sup.1(t)=IF x.sub.1(t) is
.mu..sub.1.sup.1(x.sub.1) AND x.sub.1.sub.2(t) is
.mu..sub.2.sup.1(x.sub.2) AND . . . And x.sub.m(t) is
.mu..sub.m.sup.1(x.sub.m).sub., THEN y.sub.1(t) is
.mu..sub.m+1.sup.{l.sup.m+1.sup.}(y.sub.1), y.sub.2(t) is
.mu..sub.m+2.sup.{l.sup.m+2.sup.}(y.sub.2), . . . , y.sub.n(t) is
.mu..sub.m+n.sup.{l.sup.m+n.sup.}(y.sub.n) Where:
m is the number of inputs;
n is the number of outputs;
x.sub.i(t), i=1, . . . , m are input signals;
y.sub.j(t), j=1, . . . , n are output signals;
.mu..sub.k.sup.l.sup.k are membership functions of linguistic
variables;
k=1, . . . , m+n are the indexes of linguistic variables;
l.sub.k=2,3, . . . are the numbers of the membership functions of
each linguistic variable;
.mu..sub.k.sup.{l.sup.k.sup.}--are membership functions of output
linguistic variables, upper index; {l.sub.k} means the selection of
one of the possible indexes; and
t is a time stamp.
Consider the antecedent part of the rule: R.sub.IN.sup.1(t)=IF
x.sub.1(t) is .mu..sub.1.sup.1(x.sub.1) AND x.sub.l.sub.2(t) is
.mu..sub.2.sup.1(x.sub.2) AND . . . AND x.sub.m(t) is
.mu..sub.m.sup.1(x.sub.m) The firing strength of the rule R.sup.1
in the moment t is calculated as follows:
R.sub.fs.sup.1(t)=min[.mu..sub.1.sup.1(x.sub.1(t)),
.mu..sub.2.sup.1(x.sub.2(t)), . . . , .mu..sub.m.sup.1(x.sub.m(t))]
for the case of the min-max fuzzy inference, and as
R.sub.fs.sup.1(t)=.PI.[.mu..sub.1.sup.1(x.sub.1(t)),
.mu..sub.2.sup.1(x.sub.2(t)), . . . , .mu..sub.m.sup.1(x.sub.m(t))]
for the case of product-max fuzzy inference.
In general case, here can be used any of the T-norm operations.
The total firing strength R.sub.fs.sup.1 of the rule, the quantity
R.sub.fs.sup.1(t) can be calculated as follows:
.times..intg..times..function..times..times.d ##EQU00011## for a
continuous case, and:
.times..times..times..function..times. ##EQU00012## for a discrete
case.
In a similar manner the firing strength of each s-th rule is
calculated as:
.times..intg..times..function..times..times.d.times..times..times..times.-
.function. ##EQU00013## where
.times..times. ##EQU00014##
is a linear rule index
N--number of points in the teaching signal or maximum of t in
continuous case.
In one embodiment the local firing strength of the rule can be
calculated in this case instead of integration, the maximum
operation is taken in equation (6.3):
.times..function. ##EQU00015##
In this case, the total strength of all rules will be:
.times. ##EQU00016## where:
.times..times..times..times..times..times..times..times..times..times..ti-
mes..times..times. ##EQU00017##
Quantity R.sub.fs is important since it shows in a single value the
integral characteristic of the rule base. This value can be used as
a fitness function which optimizes the shape parameters of the
membership functions of the input linguistic variables, and its
maximum guaranties that antecedent part of the KB describes well
the mutual behavior of the input signals. Note that this quantity
coincides with the "horizontal relations," introduced in the
previous section, thus it is optimized automatically by GA1.
Alternatively, if the structure of the input membership functions
is already fixed, the quantities R.sub.fs.sup.s can be used for
selection of the certain number of fuzzy rules. Many hardware
implementations of FCs have limits that constrain, in one
embodiment, the total possible number of rules. In this case,
knowing the hardware limit L of a certain hardware implementation
of the FC, the algorithm can select L.ltoreq.L.sub.0 of rules
according to a descending order of the quantities R.sub.fs.sup.s.
Rules with zero firing strength can be omitted.
It is generally advantageous to calculate the history of membership
functions activation prior to the calculation of the rule firing
strength, since the same fuzzy sets are participating in different
rules. In order to reduce the total computational complexity, the
membership function calculation is called in the moment t only if
its argument x(t) is within its support. For Gaussian-type
membership functions, support can be taken as the square root of
the variance value .sigma..sup.2.
In one embodiment, a rule pre-selection algorithm is based on a
firing strength of the rules. The threshold level can be selected
based on the maximum number of rules desired, based on user inputs,
based on statistical data and/or based on other considerations.
Rules with relatively high firing strength will be kept, and the
remaining rules are eliminated. Rules with zero firing strength can
be eliminated by default. In one embodiment, the presence of the
rules with zero firing strength may indicate the explicitness of
the linguistic variables (linguistic variables contain too many
membership functions). The total number of the rules with zero
firing strength can be reduced during membership functions
construction of the input variables. This minimization is equal to
the minimization of the "vertical relations."
This algorithm produces an optimal configuration of the antecedent
part of the rules prior to the optimization of the rules.
Optimization of the consequential part of KB can be applied
directly to the optimal rules only, without unnecessary
calculations of the "un-optimal rules". This process can also be
used to define a search space for the GA (GA2), which finds the
output (consequential) part of the rule.
A chromosome for the GA2 which specifies the structure of the
output part of the rules can be defined as: [I.sub.1, . . . ,
I.sub.M], I.sub.i=[I.sub.1, . . . , I.sub.n], I.sub.k={1, . . . ,
l.sub.Y.sub.k}, k=1, . . . , n where:
I.sub.i are groups of genes which code single rule;
I.sub.k are indexes of the membership functions of the output
variables;
n is the number of outputs; and
M is the number of rules.
In one embodiment the history of the activation of the rules can be
associated with the history of the activations of membership
functions of output variables or with some intervals of the output
signal in the Sugeno fuzzy inference case. Thus, it is possible to
define which output membership functions can possibly be activated
by the certain rule. This allows reduction of the alphabet for the
indexes of the output variable membership functions from {{1, . . .
, l.sub.Y.sub.1}, . . . , {1, . . . , l.sub.Y.sub.n}}.sup.N to the
exact definition of the search space of each rule:
{l.sup.min.sub.Y.sub.1, . . . , l.sup.max.sub.Y.sub.1}.sub.1, . . .
. , {l.sup.min.sub.Y.sub.n, . . . , l.sup.max.sub.Y.sub.n}.sub.l, .
. . . , {l.sup.min.sub.Y.sub.1, . . . ,
l.sup.max.sub.Y.sub.1}.sub.N, . . . , {l.sup.min.sub.Y.sub.n, . . .
, l.sup.max .sub.Y.sub.n}.sub.N
Thus the total search space of the GA is reduced. In cases where
only one output membership function is activated by some rule, such
a rule can be defined automatically, without GA2 optimization.
In one embodiment in case of Sugeno 0 order FIS, instead of indexes
of output membership functions, corresponding intervals of the
output signals can be taken as a search space.
For some combinations of the input-output pairs of the teaching
signal, the same rules and the same membership functions are
activated. Such combinations are uninteresting from the rule
optimization view point, and hence can be removed from the teaching
signal, reducing the number of input-output pairs, and as a result
total number of calculations. The total number of points in the
teaching signal (t) in this case will be equal to the number of
rules plus the number of conflicting points (points when the same
inputs result in different output values).
The previous section described optimization of the FIS, without the
details into the type of FIS selection. In one embodiment, the
fitness function used in the GA2 depends, at least in part, on the
type of the optimized FIS. Examples of fitness functions for the
Mamdani, Sugeno and/or Tsukamoto FIS models are described herein.
One of ordinary skill in the art will recognize that other fuzzy
models can be used as well.
Define error E.sup.p as a difference between the output part of
teaching signal and the FIS output as:
.times..function..times..times. ##EQU00018## where x.sub.1.sup.p,
x.sub.2.sup.p, . . . , x.sub.n.sup.p and d.sup.p are values of
input and output variables in the p training pair, respectively.
The function F(x.sub.1.sup.p, x.sub.2.sup.p, . . . , x.sub.n.sup.p)
is defined according to the chosen FIS model.
For the Mamdani model, the function F(x.sub.1.sup.p, x.sub.2.sup.p,
. . . , x.sub.n.sup.p) is defined as:
.function..times..times..times..times..times..mu..function..times..times.-
.mu..function..times..times..times. ##EQU00019## where
.times..mu..function. ##EQU00020## is the point of maximum value
(called also as a central value) of .mu..sub.y.sup.l(y), .PI.
denotes the selected T-norm operation.
Typical rules in the Sugeno fuzzy model can be expressed as
follows:
IF x.sub.1 is .mu..sup.(l).sub.j.sub.1(x.sub.1) AND x.sub.2 is
.mu..sup.(l).sub.j.sub.2(x.sub.2) AND . . . AND x.sub.n is
.mu..sup.(l).sub.j.sub.n(x.sub.n)
THEN y=f.sup.l(x.sub.1, . . . , x.sub.n),
where l=1,2, . . . , M--the number of fuzzy rules M defined as
{number of membership functions of x.sub.1 input
variable}.times.{number of membership functions of x.sub.2 input
variable}.times. . . . .times.{number of membership functions of
x.sub.n input variable}.
The output of Sugeno FIS is calculated as follows:
.function..times..times..times..times..mu..function..times..times..mu..fu-
nction. ##EQU00021##
Typical rules in the first-order Sugeno fuzzy model can be
expressed as follows:
IF x.sub.1 is .mu..sup.(l).sub.j.sub.1(x.sub.1) AND x.sub.2 is
.mu..sup.(l).sub.j.sub.2(x.sub.2) AND . . . AND x.sub.n is
.mu..sup.(l).sub.j.sub.n(x.sub.n)
THEN y=f.sup.l(x.sub.1, . . . ,
x.sub.n)=p.sub.1.sup.(l)x.sub.1+p.sub.2.sup.(l)x.sub.2+ . . .
p.sub.n.sup.(l) x.sub.n+r.sup.(l),
(Output variables described by some polynomial functions.)
The output of Sugeno FIS is calculated according equation
(6.6).
Typical rules in the zero-order Sugeno FIS can be expressed as
follows:
IF x.sub.1 is .mu..sup.(l).sub.j.sub.1(x.sub.1) AND x.sub.2 is
.mu..sup.(l).sub.j.sub.2(x.sub.2) AND . . . AND x.sub.n is
.mu..sup.(l).sub.j.sub.n(x.sub.n)
THEN y=r.sup.(l),
The output of zero-order Sugeno FIS is calculated as follows
.function..times..times..times..times..mu..function..times..times..mu..fu-
nction. ##EQU00022##
The typical rule in the Tsukamoto FIS is:
IF x.sub.1 is .mu..sup.(l).sub.j.sub.1(x.sub.1) AND x.sub.2 is
.mu..sup.(l).sub.j.sub.2(x.sub.2) AND . . . AND x.sub.n is
.mu..sup.(l).sub.j.sub.n(x.sub.n)
THEN y is .mu..sup.(l).sub.k(y),
where j.sub.1.di-elect cons.I.sub.m.sub.1 is the set of membership
functions describing linguistic values of x.sub.1 input variable;
j.sub.2.di-elect cons.I.sub.m.sub.2 is the set of membership
functions describing linguistic values of x.sub.2 input variable;
and so on, j.sub.n.di-elect cons.I.sub.m.sub.n is the set of
membership functions describing linguistic values of x.sub.n input
variable; and k.di-elect cons.O is the set of monotonic membership
functions describing linguistic values of y output variable.
The output of the Tsukamoto FIS is calculated as follows:
.function..times..times..times..times..mu..function..times..times..mu..fu-
nction..times..times..times..times..times..times..mu..function..times..tim-
es..times..times..mu..function. ##EQU00023##
Stage 4 described above generates a KB with required robustness and
performance for many practical control system design applications.
If performance of the KB generated in Stage 4 is, for some reasons,
insufficient, then the KB refinement algorithm of Stage 5 can be
applied.
In one embodiment, the Stage 5 refinement process of the KB
structure is realized as another GA (GA3), with the search space
from the parameters of the linguistic variables. In one embodiment
the chromosome of GA3 can have the following structure:
{[.DELTA..sub.1,.DELTA..sub.2,.DELTA..sub.3]}.sup.L;
.DELTA..sub.i.di-elect cons.[-prm.sub.i.sup.j,1-prm.sub.i.sup.j];
i=1,2,3;j=1,2, . . . , L, where L is the total number of the
membership functions in the system
In this case the quantities .DELTA..sub.i are modifiers of the
parameters of the corresponding fuzzy set, and the GA3 finds these
modifiers according to the fitness function as a minimum of the
fuzzy inference error. In such an embodiment, the refined KB has
the parameters of the membership functions obtained from the
original KB parameters by adding the modifiers
prm.sup.new.sub.i=prm.sub.i+.DELTA..sub.i.
Different fuzzy membership function can have the same number of
parameters, for example Gaussian membership functions have two
parameters, as a modal value and variance. Iso-scalene triangular
membership functions also have two parameters. In this case, it is
advantageous to introduce classification of the membership
functions regarding the number of parameters, and to introduce to
GA3 the possibility to modify not only parameters of the membership
functions, but also the type of the membership functions, form the
same class.
GA3 improves fuzzy inference quality in terms of the approximation
error, but may cause over learning, making the KB too sensitive to
the input. In one embodiment a fitness function for rule base
optimization is used. In one embodiment, an information-based
fitness function is used. In another embodiment the fitness
function used for membership function optimization in GA1 is used.
To reduce the search space, the refinement algorithm can be applied
only to some selected parameters of the KB. In one embodiment
refinement algorithm can be applied to selected linguistic
variables only.
The structure realizing evaluation procedure of GA2 or GA3 is shown
in FIG. 12B. In FIG. 12B, the SC optimizer 12001 sends the KB
structure presented in the current chromosome of GA2 or of GA3 to
FC 12101. An input part of the teaching signal 12102 is provided to
the input of the FC 12101. The output part of the teaching signal
is provided to the positive input of adder 12103. An output of the
FC 12101 is provided to the negative input of adder 12103. The
output of adder 12103 is provided to the evaluation function
calculation block 12104. Output of evaluation function calculation
block 12104 is provided to a fitness function input of the SC
optimizer 12001, where an evaluation value is assigned to the
current chromosome.
In one embodiment evaluation function calculation block 12104
calculates approximation error as a weighted sum of the outputs of
the adder 12103.
In one embodiment evaluation function calculation block 12104
calculates the information entropy of the normalized approximation
error.
In one embodiment of Stages 4 and 5 the fitness function of GA can
be represented as some external function Fitness=f(KB), which
accepts as a parameter the KB and as output provides KB
performance. In one embodiment, the function f includes the model
of an actual plant controlled by the system with FC. In this
embodiment, the plant model in addition to plant dynamics provides
for the evaluation function.
In one embodiment function f might be an actual plant controlled by
an adaptive LINEAR controller with coefficient gains scheduled by
FC and measurement system provides as an output some performance
index of the KB.
In one embodiment the output of the plant provides data for
calculation of the entropy production rate of the plant and of the
control system while the plant is controlled by the FC with the
structure from the KB.
In one embodiment, the evaluation function is not necessarily
related to the mechanical characteristics of the motion of the
plant (such as, for example, in one embodiment control error) but
it may reflect requirements from the other viewpoints such as, for
example, entropy produced by the system, or harshness and or bad
feelings of the operator expressed in terms of the frequency
characteristics of the plant dynamic motion and so on.
FIG. 12C shows one embodiment the structure-realizing KB evaluation
system based on plant dynamics. In FIG. 12C, and SC optimizer 12201
provides the KB structure presented in the current chromosome of
the GA2 or of the GA3 to an FC 12301. the FC is embedded into the
KB evaluation system based on plant dynamics 12300. The KB
evaluation system based on plant dynamics 12300 includes the FC
12301, an adaptive LINEAR controller 12302 which uses the FC 12301
as a scheduler of the coefficient gains, a plant 12303, a
stochastic excitation generation system 12304, a measurement system
12305, an adder 12306, and an evaluation function calculation block
12307. An output of the linear controller 12302 is provided as a
control force to the plant 12303 and as a first input to the
evaluation function calculation block 12307. Output of the
excitation generation system 12304 is provided to the Plant 12303
to simulate an operational environment. An output of the Plant
12303 is provided to the measurement system 12305. An output of the
measurement system 12305 is provided to the negative input of the
adder 12306 and together with the reference input Xref forms in
adder 12306 control error which is provided as an input to the
linear controller 12302 and to the FC 12301. An output of the
measurement system 12305 is provided as a second input of the
evaluation function calculation block 12307. The evaluation
function calculation block 12307 forms the evaluation function of
the KB and provides it to the fitness function input of SC
optimizer 12201. Fitness function block of SC optimizer 12201 ranks
the evaluation value of the KB presented in the current chromosome
into the fitness scale according to the current parameters of the
GA2 or of the GA3.
In one embodiment, the evaluation function calculation block 12307
forms evaluation function as a minimum of the entropy production
rate of the plant 12303 and of the linear controller 12302.
In one embodiment, the evaluation function calculation block 12307
applies Fast Fourier Transformation on one or more outputs of the
measurement system 12305, to extract one or more frequency
characteristics of the plant output for the evaluation.
In one embodiment, the KB evaluation system based on plant dynamics
12300 uses a nonlinear model of the plant 12303. In one embodiment,
the KB evaluation system based on plant dynamics 12300 is realized
as an actual plant with one or more parameters controlled by the
adaptive linear controller 12302 with control gains scheduled by
the FC 12301. In one embodiment plant 12303 is a stable plant. In
one embodiment plant 12303 is an unstable plant. The output of the
SC optimizer 12201 is an optimal KB 12202.
In FIG. 13, the structure of the fuzzy controller system and its
development stages is described, combining the following Stages:
Stage 1 (13100), teaching signal acquisition stage for several
initial conditions; Stage 2 (13200), arbitrary course
simulation.
In stage 1 (13100), several combinations of reference velocities
and roll angles are set, which are also used as initial velocities
and roll angles. The plant model of motorcycle 13101 (as shown in
FIG. 3) and PD controller for steering 13103 and P controller for
driving 13104 are embedded into the structure of Stage 1 (13100).
The calculation for controllers 13103, 13104 is based on the
equation (7-1)-(7-2). And the gain parameters for the PD controller
for steering 13103 and the P controller for driving 13104 are
provided by the GA 13102. The GA 13102 provides optimization of the
parameters executed through the evaluation of the value of the
Fitness Function 13105 for various conditions. The calculation of
the Fitness Function 13105 is based on the equation (7-6).
Optimized parameters are stored in the KB 13002. Data for the KB
13002 are transformed into fuzzy rule form for the KB 13003 by the
SCO 13001. The same plant model of MC 13201 and PD controller for
steering 13203 and P controller for driving 13204 are embedded into
the structure of Stage 2 (13200). The calculation for the
controllers 13203, 13204 is based on equations (7-1)-(7-2). In this
stage, the gain parameters for the PD controller for steering 13203
and P controller for driving 13204 are provided by the FC 13206
using the KB of the fuzzy rule 13003. Input data for the P
controller for driving 13204 is the difference between forward
velocity and reference velocity. Input data for the PD controller
for steering 13203 are the difference between roll angle and
reference roll angle, which is calculated by .phi..sub.ref
calculator 13207. Input data for the calculator 13207 are LR 13208,
course data 13212 and position and yaw of the plant model of the MC
13201. The calculation is based on equations (7-3)-(7-5).
.tau..function..times..times..tau..function..PHI..PHI..times..PHI..times.-
.times..times..times..times..times..times..times..times..PHI..function..ti-
mes..times..times..times..times. ##EQU00024##
FF=(v-v.sub.ref).sup.2.times.0.2+(.phi.-.phi..sub.ref).sup.2+{dot
over
(.phi.)}.sup.2+.tau..sub.d.sup.20.0001+.tau..sub.s.sup.2.times.0.0005
(7-6)
FIG. 14 is a block diagram showing the structure of the hold
steering torque feedforward system and its development. FIG. 14
shows the following 3 stages. Stage 1 (14100) is a data acquisition
stage for several conditions. Stage 2 (14200) is a parameter
optimization stage for several forward velocities. Stage 3 (14300)
is an arbitrary course simulation.
In Stage 1 (14100), several reference velocities are set. The
reference velocities are also used as initial velocities. Several
radius of circle course 14112 and several LR 14108 are used. As
combination of several reference velocities and several radius of
circle course 14112 and several LR 14108, many cases of simulation
must be executed. The plant model of the motorcycle 14101, the PD
controller for steering 14103, and the P controller for driving
14104 are embedded into the structure of Stage 1 (14100). The
calculation for controllers 14103, 14104 is based on equation
(7-1)-(7-2). The P gain parameter for PD controller for steering
14103 and P controller for driving 14104 are fixed. The D parameter
for PD controller for steering 14103 are provided by a GA 14102.
The GA 14102 feeds the value of .phi..sub.col 14109 and
.tau..sub.hold 14110. .phi..sub.col 14109 is used as input for
.phi..sub.ref calc. 14107. Other input data of the calculator 14107
are LR 14108 and course data (a circular course is used here) 14112
and position and yaw of the plant model of the motorcycle 14101.
The calculation 14107 is based on equations (7-7)-(7-111). The
value .tau..sub.hold 14110 is added to the output of the PD
controller for steering 14103. By using the GA 14102, the
optimization of the parameters is executed through the evaluation
of the value of the Fitness Function 14105 for each desired
condition. The calculation of Fitness Function 14105 is based on
equation (7-12). Optimized parameters of .phi..sub.col 14109 and
.tau..sub.hold 14110 are stored in the KB 14001.
The data of KB 14001 are used in the Look up table 14211 in Stage 2
(14200). The Look up table 14211 brings optimized .phi..sub.col
14209 and .tau..sub.hold 14210 according to LR 14208 and reference
roll angle calculated by .phi..sub.ref calc. 14107 and forward
velocity of the plant model of the motorcycle 14201. In stage 2
(14200), several reference velocities are set, which are also used
as initial velocities. The plant model of the motorcycle 14201, the
PD controller for steering 14203, and the P controller for driving
14204 are embedded into the structure of Stage 2 (14200). The
calculation for controllers 14203, 14204 is based on equations
(7-1)-(7-2). The gain parameters for P controller for driving 14204
are fixed. In this stage, the gain parameters for the PD controller
for steering 14203 are provided by a GA 14202. The LR 14208 is also
provided by the GA 14202. The LR 14208, course data (lane change
course which is shown in FIG. 30(a) is used here) 14212 and
position and yaw of the plant model of motorcycle 14201 and
optimized .phi..sub.col 14209 are used as inputs for .phi..sub.ref
calc 14207. The calculation 14207 is based on equations
(7-7)-(7-11). The value .tau..sub.hold 14210 is added to the output
of the PD controller for steering 14203. By using the GA 14202, the
optimization of the parameters is executed through the evaluation
of the value of the Fitness Function 14205 for each reference
velocity. The calculation of the Fitness Function 14205 is based on
equations (7-13). Optimized parameters of the gain parameters for
the PD controller for steering 14203 and LR 14208 are added to the
KB 14001, and the KB 14002 is constructed.
The data of the KB 14002 are used in the Look-up table 14311 in
Stage 3 (14300). The Look-up table 14311 brings optimized
.phi..sub.col 14309 and .tau..sub.hold 14310 and the gain
parameters for the PD controller for steering 14303 and LR 14308
according to reference roll angle calculated by .phi..sub.ref calc
14107 and forward velocity of the plant model of motorcycle 14301.
The plant model of the motorcycle 14301, the PD controller for
steering 14303, and the P controller for driving 14304 are embedded
into the structure of Stage 3 (14300). The calculation for
controllers 14303, 14304 is based on equations (7-1)-(7-2).
The gain parameters for the P controller for driving 14304 are
fixed. In this stage, the gain parameters for the PD controller for
steering 14303 are provided by the Look-up table 14311 as mentioned
above. The LR 14308, arbitrary course data 14312, and the position
and yaw of the plant model of the motorcycle 14301 and optimized
.phi..sub.col 14309 are used as inputs for .phi..sub.ref calc
14307. The calculation 14307 is based on equation (7-7)-(7-11). The
value .tau..sub.hold 14310 is added to the output of the PD
controller for steering 14303.
.omega..times..times..omega..times..times..times..times..times..PHI..time-
s..times..function..function..times..times..times..PHI..PHI..times..times.-
.PHI..times..times..PHI..PHI..times..times..tau..times..times..times.
##EQU00025##
The optimized KB defines the optimal fuzzy control system to
realize a robust rider model, which can drive the motorcycle (plant
model) passing through an arbitrary course trace with arbitrary
velocity without fall down, and within the desired control error
tolerance and control quality.
In one embodiment, the rider model presented is a combination model
of look-forward a model and an ideal roll control model. To
calculate steering torque, using deviation at length of reference
is from look-forward model and using roll rate is from ideal roll
control model (see FIG. 15). The rider model uses a fuzzy
controller that optimizes the gain parameters corresponding to the
velocity and the roll angle. To optimize the gain parameters, the
Soft Computing Optimizer based on GA (genetic algorithm) with
different types of fitness function is used in a simulation to
control the velocity and roll angle to reference value. The method
using the Soft Computing Optimizer is divided into two parts. The
first part is to gain the teaching signals by simulation, and the
second part is to make the optimized fuzzy controller by using the
teaching signals.
The equations for controller torque are:
.tau..sub.d=-K.sub.p1(v-v.sub.ref) (9-1)
.tau..sub.s=-K.sub.p2(.phi.-.phi..sub.ref)-K.sub.d2.phi. (9-2)
The equations for reference roll angle are:
.times..times..times..times..times..PHI..function..function..times..times-
..times. ##EQU00026##
The equations for steer torque and body movement controller are:
.tau..sub.s=K.sub..phi.s.phi.+K.sub.ysy (9-6)
.tau..sub.1=K.sub..tau..tau..sub.s (9-7)
.tau..sub.2=K.sub..phi.2.phi.+K.sub.y2y (9-8)
The reference roll angle is given by course radius:
.tau..sub.d=K.sub.p1(v.sub.ref-v) (9-9)
.tau..sub.s=K.sub.p(.phi..sub.ref-.phi.)+K.sub.l.intg.(.phi..sub.ref-.phi-
.)dt+K.sub.D(.phi..sub.ref-.phi.) (9-10)
To obtain the teaching signal, the Soft Computing Optimizer uses a
program mexGA, a MATLAB program script.m a MATLAB program fitness.m
and the MC model. The structure is shown in FIG. 16. In script.m
the initial conditions and reference values are set. In fitness.m
the fitness function is defined as: M-file: file in MATLAB language
Reference velocity=5 7 10 15 m/s Reference roll angle=0.degree.
10.degree. 30.degree. Optimized gain Kp1, Kp2, Kd2 Fitness function
FF=(v-v.sub.ref).sup.2.times.0.2+(.phi.-.phi..sub.ref).sup.2+{dot
over
(.phi.)}.sup.2+.tau..sub.d.sup.2.times.0.0001+.tau..sub.s.sup.2=0.0005
(7-6) GA parameter Population=100 Generation=5 Mutation_rate=0.6
Crossover_rate=0.9 Delete_rate=0.8 Fitness_reduce=0.05
FIG. 17 shows examples of teaching signals obtained from the
simulation. FIG. 18 shows results of optimized parameters for
different initial conditions. Several features of the teaching
signals produced by simulation are noted as follows. Kp1 is
relatively large when the velocity is fast, regardless of roll
angle. Kp2 change relatively little except when velocity is slow
and roll angle is large. When the reference roll angle is 30 deg.
and the reference velocity is 5 m/s, the resulting roll angle
converges at less than 27 deg., regardless of gain parameters. This
appears to be the basis for the preceding exception. Torque of the
front tire around the steering axis is largely the reason for this
phenomenon. Kd2 is relatively large when the velocity is slow,
regardless of roll angle.
As a next step, the SC optimizer computes the optimum structure of
the KB for the fuzzy controller. The input for the SC optimizer is
the teaching signal. The membership functions for the KB are set to
make the input data of fuzzy operation from the input data of the
fuzzy controller, such as velocity and roll angle. The SC optimizer
determines functions and makes fuzzy rules and optimizes them by
the teaching signals using GA. FIGS. 19 and 20 shows the results of
membership function design and FC output, correspondingly.
In FIG. 21 is a block diagram of the simulation model including a
fuzzy controller block 2101 The block 2102 serves as the external
performance function of the SC optimizer, which provides gain
parameters for the simulation using the fuzzy rules. A reference
roll angle calculator 2103 block calculates the reference roll
angle from assigned course data and LR (length of reference) and
input data (position, velocity and yaw).
The test course assigned here and the simulation are shown in FIG.
22. The course is driven in a counterclockwise direction. The
course starts in a straight line and goes into a J-turn of 13 meter
radius. After a short straight and turn, it goes to lane change and
J-turn of 10 meter radius. The course then goes through a large
radius turn and back the starting straight. In this simulation, the
motorcycle run along this course two times, and as the reference
velocity the experiment data (FIG. 23A) was used. In the first turn
the reference velocity is approximately 5 m/s and in the second
turn the reference velocity is approximately 7 m/s. LR is 6 meters
for the first turn and is 7.5 meters for the second turn.
FIGS. 23B-E show the resulting data of simulation and corresponding
measured data. At the 10 meter J-turn R10 in the second circle (in
time scale 70 sec), the simulation data vibrates strongly compared
to experiment data. The reason lies in the torque of the front tire
around the steering axis. When it is about to go into a regular
circle, the calculated control torque approaches 0. If the torque
of the front tire around the steering axis is 0, the regular circle
revolution would be kept. But the torque is not 0, so the vibration
is provoked. FIG. 23F shows the change of gain parameters by the
fuzzy controller.
To evaluate the robustness of the controller, a simulation with
disturbance in tire load was carried out as shown in FIGS. 24A-E.
The effect is relatively small as the frequency of the weave mode,
which is the dominant mode of MC at researched velocity range, is
about 0.5 Hz. So the high frequency noise effect is small. If the
velocity range was different and the high frequency modes were
dominant, the influence of the tire noise would be more
important.
The other tests for the robustness was simulations assuming the
transport delay of the actuator. As shown in FIGS. 25A-D, until
0.04 sec of delay is introduced, the result did not change much.
However, when assuming a 0.05 sec delay, the vibration of the MC
diverged and the simulation became unstable.
In optimization of control of a motorcycle, stabilizing roll angle
goes together with steering the desired course (see FIG. 26A). It
is thus useful to first explore control methods that can stabilize
the motorcycle when driven in a circle. The reason is that roll
angle will be almost constant in a circular path for constant
velocity and any course desired can be resolved into combinations
of short circular arcs. If the control method can stabilize the
motorcycle through a circular arc of any desired radius, then the
MC can be stably steered through any course constructed from arcs
having radii within the design range.
Accordingly, a circular course was used for the simulation and the
circular course was run while computing a reference roll angle, and
optimization of the parameter by the SC optimizer was used during
the simulation (see FIG. 26B). As described by equation 9-4, it is
necessary to provide a holding torque to counter the torque of the
front tire around the steering axis to hold the reference roll
angle. So, a feedforward rider model can be used. The method of
computing the reference roll angle in equations (9-3)-(9-5) assumes
the course deviation to be small. Ignoring the change of yaw, the
constant lateral acceleration cancels the predicted course
deviation. So when the course deviation is large, computing
reference roll angle by this method has a large error. The present
method for computing reference roll angle (equations (9-14)-(9-17)
assumes the arc course. So this method fits the circle course
rather better. However, there is still some difference between
computed reference roll angle and actual converged roll angle. The
correction value introduced in equation (9-18) and optimized in
simulation of circle course reduces the error. The correction
compensates for the overturning moment of the tire and wheel base
length. The optimization of parameter (the holding torque and the
correction value of reference roll angle) by simulation of circle
course, the controllability for transient response can not be
optimized enough. The lane change simulation was added for
optimization of length of reference (LR) and the PD gains (Kp2,
Kd2).
In this method velocity control gain was fixed. The equations for
controller torque are: .tau..sub.d=-K.sub.p1(v-v.sub.ref) (9-12)
.tau..sub.s=-K.sub.p2(.phi.-.phi..sub.ref)-K.sub.d2.phi.+.tau..sub.offset
(9-13) The equations for reference roll angle are:
.omega..times..times..omega..times..times..times..times..times..times..ti-
mes..times..times..times..times..PHI..times..times..function..function..ti-
mes..times..times..PHI..PHI..times..times..PHI..times..times.
##EQU00027##
FIG. 27 shows the model for circular course simulation.
To construct the feedforward rider model, simulation of the
circular course controlled by SC optimizer was done under several
conditions. The holding torque and the correction value of the
reference roll angle were optimized and the converged roll angle
were obtained in each condition. Gain parameters Kp1 and Kp2 were
fixed (Kp1=1000, Kp2=80). The gain parameter Kd2 was added to the
optimization.
The results of the simulation are shown in FIG. 28. The holding
torque and the correction value of the reference roll angle and the
converged roll angle become larger as the radius of the course
becomes smaller. LR affects the correction value of reference roll
angle. The holding torque and the correction value of reference
roll angle are nearly in proportion to the computed reference roll
angle (FIG. 29A-B).
It is difficult to optimize the controllability of transient
response by circle course. The lane change simulation (see FIG. 30)
can be used for parameters to be optimized in transient conditions.
The result of the optimization in circle course simulation was used
in lane change simulation.
ref. velocity, /
optimized parameter Kp2 Kd2
fitness function FF=y.sub.o.sup.2+.tau..sub.s.sup.2.times.0.007
(9-19)
Holding torque was computed by interpolation of input data as a
function of the computed reference roll angle. (assuming velocity
as constant) Correction of reference roll angle was computed by
interpolation of input matrix data as function of LR and computed
reference roll angle. (assuming velocity as constant).
The feedforward rider model (see FIG. 31) was made as the table
data (Table 2) which are optimized by simulations of the circle
course and the lane change course controlled by the SC optimizer.
The resulting data of simulation in the test course fit well to
experiment data (FIGS. 32A-E). At the 10 meter J-turn in the second
circuit (in time scale 70 sec), the simulation data does not
vibrate.
The tests for the robustness in the simulation are provided by
assuming the transport delay of the steering actuator. As shown in
FIGS. 33A-D, until 0.05 sec of delay is introduced, the control
result does not change substantially. Introducing a 0.07 sec delay
causes vibration of the MC and instability in the simulation.
FIG. 34 shows the sensor and measurement apparatus. The measuring
system has two portions. The first portion 34200 is integrated into
the test scooter as shown in FIG. 35. The second portion 34100 is
the measurement base which includes a wireless modem 34101, GPS
reference station 34102, transmitter board 34103, and computer with
monitor 34104. The measured data (position and attitude of the
scooter, steering angle, steering torque) can be seen on the
monitor as shown in FIG. 36. The first portion 34200 integrated in
the test scooter includes a wireless modem 34201, steering sensor
system 34202, which is installed in the steering part of the
scooter as shown in FIG. 35, driving motor sensor 34203, position
and attitude sensors 34204 attached to the scooter as shown in FIG.
35, and ECU unit 34205. The block 34202 and 34203 are the
additional part to the original system in order to specialize the
original system for the scooter measurement.
* * * * *